Question

When considering the set of all the natural numbers (ℕ), show whether the mathematical operations of addition, subtraction, multiplication and division are: (a) Associative (b) Commutative.

Answer 1

(a) Since $a+(b+c)=a+b+c=(a+b)+c$ for any $a,b,c\in\N,$ we conclude that the operation of addition is associative on the set $\N$ of natural numbers.

Since $a\cdot(b\cdot c)=a\cdot b\cdot c=(a\cdot b)\cdot c$ for any $a,b,c\in\N,$ we conclude that the operation of multiplication is associative on the set $\N$ of natural numbers.

Since $4-(2-1)=3\ne 1=(4-2)-1,$ we conclude that the operation of subtraction is not associative on the set $\N$ of natural numbers.

Since $8:(4:2)=4\ne 1=(8:4):2,$ we conclude that the operation of division is not associative on the set $\N$ of natural numbers.

(b) Since $a+b=b+a$ for any $a,b\in\N,$ we conclude that the operation of addition is commutative on the set $\N$ of natural numbers.

Since $a\cdot b=b\cdot a$ for any $a,b\in\N,$ we conclude that the operation of multiplication is commutative on the set $\N$ of natural numbers.

Since $2-1=1\ne -1=1-2,$ we conclude that the operation of subtraction is not commutative on the set $\N$ of natural numbers.

Since $8:4=2\ne 0.5=4:8,$ we conclude that the operation of division is not commutative on the set $\N$ of natural numbers.